Journal of the Indian Math. Soc. Vol. 81, Nos. 1-4, (2014), 45–59.
IMPACT OF (CLRST ) PROPERTY AND EXISTENCE OF FIXED POINTS USING IMPLICIT RELATIONS SUNNY CHAUHAN, ISMAT BEG AND B. D. PANT Abstract. In this paper, we utilize the notion of “common limit range property” and prove common fixed point theorems for weakly compatible mappings in Menger spaces satisfying implicit relations. We give some examples to support the useability of our main results. Our results improve and generalize several previously known fixed point theorems in the existing literature.
(Received: February 13, 2012, Accepted: October 26, 2012) 1. Introduction The concept of a probabilistic metric space (briefly, PM-space) was initiated and studied by Karl Menger [17]. The idea of Menger was to use distribution functions instead of non-negative real numbers as values of the metric. The notion of PM-space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. In fact the study such spaces received an impetus with the pioneering work of Schweizer and Sklar [23]. A probabilistic generalization of metric spaces appears to be of interest in the investigation of physical quantities and physiological thresholds. The theory of PM-spaces is of paramount importance in probabilistic functional analysis especially due to its extensive applications in random differential as well as random integral equations. In 1986, Jungck [12] introduced the notion of compatible mappings in metric space. A metrical common fixed point theorem generally involves conditions continuity, contraction and completeness (or closedness) of the underlying space (or subspaces) along with conditions on suitable containment amongst the ranges of involved mappings. Hence, in order to prove a new metrical common fixed point theorem, one is always required to improve one or more of 2010 Mathematics Subject Classification. Primary 47H10, Secondary 54H25. Key words and phrases: t-norm; Menger space; property (E.A); common property (E.A); (CLRS ) property; (CLRST ) property; weakly compatible mappings. c Indian Mathematical Society, ISSN 0019–5839
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these conditions. With a view to improve commutativity conditions in common fixed point theorems, Jungck and Rhoades [13] introduced the concept of weak compatibility to the setting of single-valued and multi-valued mappings which is more general than compatibility. However, the study of common fixed points of non-compatible mappings is also equally interesting which was initiated by Pant [20] in metric spaces. In 2002, Aamri and Moutawakil [1] defined a property (E.A) for self mappings which contained the class of non-compatible mappings. In 2005, Liu et al. [16] proposed the notion of common property (E.A) for a hybrid pair of single and multi-valued mappings and proved some fixed point results under hybrid contractive conditions. Researchers of this domain extended several definitions (that is, compatible mappings, noncompatible mappings, property (E.A) and common property (E.A)) in frame work of probabilistic settings (see [19, 14, 2, 3]). In recent years, many authors have proved a number of fixed point theorems in Menger spaces. To mention a few, we cite [4, 5, 6, 7] It is observed that property (E.A) and common property (E.A) require the closedness of the subspaces for the existence of fixed point. Recently, Sintunavarat and Kumam [25] coined the idea of “common limit range property” which never requires the closedness of the subspaces for the existence of fixed point. Imdad et al. [11] extended the notion of common limit range property to two pair of self mappings, called (CLRST ) property (with respect to mappings S and T ) and proved fixed point theorems in Menger spaces as well as in metric spaces. In fixed point theory implicit relations are used as a tool to find the common fixed point of involved mappings. In [22], Popa used the family F4 of implicit real functions for the existence of fixed point in topological space. Further, Mihet¸ [18], Singh and Jain [24] and Imdad and Ali [9] proved some fixed point theorems satisfying implicit relations. Afterward, Kumar and Pant [15] and Pant and Chauhan [21] proved some fixed point results in Menger spaces by using the notion of implicit functions. The aim of this paper is to prove a common fixed point theorem for two pairs of weakly compatible self mappings in Menger space by using the (CLRST ) property. Illustrative examples are also furnished to support our main results. As an application to our main result, we present a common fixed point theorem for four finite families of self mappings in Menger space.
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2. Preliminaries Definition 2.1. [23] A mapping F : R → R+ is called a distribution function if it is non-decreasing and left continuous with inf F (t) = 0 and t∈R
sup F (t) = 1. t∈R
We shall denote by ℑ the set of all distribution functions while H will always denote the specific distribution function defined by 0, if t ≤ 0; H(t) = 1, if t > 0. Definition 2.2. [23] A PM-space is an ordered pair (X, F ), where X is a nonempty set of elements and F is a mapping from X × X to ℑ, the collection of all distribution functions. The value of F at (x, y) ∈ X × X is represented by Fx,y . The functions Fx,y are assumed to satisfy the following conditions: (1) (2) (3) (4)
Fx,y (t) = 1 for all t > 0 if and only if x = y; Fx,y (0) = 0; Fx,y (t) = Fy,x (t); If Fx,y (t) = 1 and Fy,z (s) = 1 then Fx,z (t + s) = 1 for all x, y, z ∈ X and t, s > 0.
Definition 2.3. [23] A mapping △ : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (briefly, t-norm) if the following conditions are satisfied: for all a, b, c, d ∈ [0, 1] (1) (2) (3) (4)
△(a, 1) = a for all a ∈ [0, 1]; △(a, b) = △(b, a); △(a, b) ≤ △(c, d) for a ≤ c, b ≤ d; △(△(a, b), c) = △(a, △(b, c)).
Example 2.1. The following are the four basic t-norms: (1) (2) (3) (4)
The The The The
minimum t-norm: △M (a, b) = min{a, b}. product t-norm: △P (a, b) = a.b. Lukasiewicz t-norm: △L (a, b) = max{a + b − 1, 0}. weakest t-norm, the drastic product: min{a, b}, if max{a, b} = 1; △D (a, b) = 0, otherwise.
In respect of above mentioned t-norms, we have the following ordering: △D (a, b) < △L (a, b) < △P (a, b) < △M (a, b).
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Throughout this paper, △ stands for an arbitrary continuous t-norm. Definition 2.4. [23] A Menger space is a triplet (X, F , △) where (X, F ) is a PM-space and t-norm △ is such that the inequality Fx,z (t + s) ≥ △ (Fx,y (t), Fy,z (s)) , holds for all x, y, z ∈ X and all t, s > 0. Every metric space (X, d) can be realized as a PM-space by taking F : X × X → ℑ defined by Fx,y (t) = H(t − d(x, y)) for all x, y ∈ X. Definition 2.5. [19] A pair (A, S) of self mappings of a Menger space (X, F , △) is said to be compatible if and only if FASxn ,SAxn (t) → 1 for all t > 0, whenever {xn } is a sequence in X such that Axn , Sxn → z for some z ∈ X as n → ∞. Definition 2.6. [13] A pair (A, S) of self mappings of a non-empty set X is said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, that is, if Az = Sz (for z ∈ X), then ASz = SAz. Two compatible self mappings are weakly compatible, but the converse is not true. Definition 2.7. [14] A pair (A, S) of self mappings of a Menger space (X, F , △) is said to satisfy the property (E.A), if there exists a sequence {xn } in X for some z ∈ X such that lim Axn = lim Sxn = z.
n→∞
n→∞
Definition 2.8. [2] A pair (A, S) of self mappings of a Menger space (X, F , △) is said to be non-compatible if and only if there exists at least one sequence {xn } in X such that lim Axn = lim Sxn = z for some z ∈ X, but for some t > 0, n→∞
n→∞
lim FASxn ,SAxn (t) is either less than 1 or nonexistent.
n→∞
Remark 2.1. In view of Definition 2.7, it is straight forward to notice that every pair of non-compatible self mappings of a Menger space (X, F , △) satisfies the property (E.A) but not conversely (see [7, Example 1]). Definition 2.9. [2] Two pairs (A, S) and (B, T ) of self mappings of a Menger space (X, F , △) are said to satisfy the common property (E.A), if there exist two sequences {xn }, {yn} in X for some z in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z.
n→∞
n→∞
n→∞
n→∞
IMPACT OF (CLRST ) PROPERTY
49
Example 2.2. Let (X, F , △) be a Menger space with X = [−1, 1] and t , if t > 0; Fx,y (t) = t+|x−y| 0, if t = 0,
for all x, y ∈ X. Define the self mappings A, B, S and T on X as Ax = x, Bx = −x, Sx = x3 and T x = − x3 for all x ∈ X. Then with the sequences {xn } = n1 and {yn } = − n1 in X, one can easily verify that lim Axn = lim Sxn = lim Byn = lim T yn = 0 ∈ X.
n→∞
n→∞
n→∞
n→∞
Therefore both the pairs (A, S) and (B, T ) share the common property (E.A). Definition 2.10. [11] A pair (A, S) of self mappings of a Menger space (X, F , △) is said to satisfy the (CLRS ) property with respect to mapping S if there exists a sequence {xn } in X such that lim Axn = lim Sxn = z,
n→∞
n→∞
where z ∈ S(X). Now, we present examples of self mappings A and S satisfying the (CLRS ) property. Example 2.3. Let (X, F , △) be a Menger space with X = [0, ∞) and t , if t > 0; Fx,y (t) = t+|x−y| 0, if t = 0
for all x, y ∈ X. Define self mappings A and S on X by A(x) = x + 6 and S(x) = 6x for all x ∈ X. Let a sequence {xn } = {1 + n1 }n∈N in X, we have lim Axn = lim Sxn = 6 = S(1),
n→∞
n→∞
that is, 6 ∈ S(X), which shows that the pair (A, S) enjoys the (CLRS ) property. Example 2.4. The conclusion of Example 2.3 remains true if the self mappings A and S are defined on X by A(x) = x2 and S(x) = 2x 3 for all x ∈ X. Let a sequence {xn } = { n1 }n∈N in X. Since lim Axn = lim Sxn = 0 = S(0),
n→∞
n→∞
that is, 0 ∈ S(X), hence A and S satisfy the (CLRS ) property.
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Remark 2.2. From the Examples 2.3-2.4, it is evident that a pair (A, S) satisfying the (E.A) property along with closedness of the subspace S(X) always enjoys the (CLRS ) property. Definition 2.11. [11] Two pairs (A, S) and (B, T ) of self mappings of a Menger space (X, F , △) are said to satisfy the (CLRST ) property (with respect to mappings S and T ) if there exist two sequences {xn }, {yn } in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z, n→∞
n→∞
n→∞
n→∞
where z ∈ S(X) ∩ T (X). n Definition 2.12. [10] Two families of self mappings {Ai }m i=1 and {Bk }k=1 are said to be pairwise commuting if
(1) Ai Aj = Aj Ai for all i, j ∈ {1, 2, . . . , m}, (2) Bk Bl = Bl Bk for all k, l ∈ {1, 2, . . . , n}, (3) Ai Bk = Bk Ai for all i ∈ {1, 2, . . . , m} and k ∈ {1, 2, . . . , n}. 3. Implicit Relations Following by Singh and Jain [24], let Φ be the class of all real continuous functions ϕ : (R+ )4 → R, non-decreasing in the first argument and satisfying the following conditions: (ϕ1 ): u, v ≥ 0, ϕ(u, v, u, v) ≥ 0 or ϕ(u, v, v, u) ≥ 0 implies that u ≥ v, (ϕ2 ): ϕ(u, u, 1, 1) ≥ 0 for all u ≥ 1. Example 3.1. [21] Define ϕ(t1 , t2 , t3 , t4 ) = at1 +bt2 +ct3 +dt4 , where a, b, c, d ∈ R with a + b + c + d = 0, a > 0, a + c > 0, a + b > 0 and a + d > 0 . Then ϕ ∈ Φ. Example 3.2. [21] Define ϕ(t1 , t2 , t3 , t4 ) = 14t1 −12t2 +6t3 −8t4 . Then ϕ ∈ Φ. Further, Imdad and Ali [9] used the following implicit relation for the existence of common fixed points of the involved mappings. Let Ψ be the class of all real continuous functions ψ : [0, 1]4 → R satisfying the following conditions: (ψ1 ): For every u > 0, v ≥ 0 with ψ(u, v, u, v) ≥ 0 or ψ(u, v, v, u) ≥ 0, we have u > v. (ψ2 ): ψ(u, u, 1, 1) < 0, for each 0 < u < 1. Example 3.3. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t1 −φ (min{t2 , t3 , t4 }), where φ : [0, 1] → [0, 1] is a continuous function such that φ(s) > s for 0 < s < 1.
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Example 3.4. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t1 − a min{t2 , t3 , t4 }, where a > 1. Example 3.5. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t1 − at2 − min{t3 , t4 }, where a > 0. Example 3.6. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t1 − at2 − bt3 − ct4 , where a > 1, b, c ≥ 0(6= 1). Example 3.7. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t1 − at2 − b(t3 + t4 ), where a > 1, b ≥ 0(6= 1). Example 3.8. Define ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = t31 − at2 t3 t4 , where a > 1. Here, it may be noted that above mentioned classes of functions Φ and Ψ are independent. For more details, we refer to [8]. 4. Results Before proving our main theorems, we begin with the following observation. Lemma 4.1. Let A, B, S and T be self maps of a Menger space (X, F , △) satisfying the following conditions: (1) the pair (A, S) satisfies the (CLRS ) property or the pair (B, T ) sat isfies the (CLRT ) property , (2) A(X) ⊂ T (X) or B(X) ⊂ S(X) , (3) T (X) or S(X) is a closed subset of X, (4) B(yn ) converges for every sequence {yn } in X whenever T (yn ) con verges
or A(xn ) converges for every sequence {xn } in X whenever S(xn ) converges , (5) there exists ψ ∈ Ψ such that ψ (FAx,By (t), FSx,T y (t), FAx,Sx (t), FBy,T y (t)) ≥ 0,
(4.1)
for all x, y ∈ X and t > 0. Then the pairs (A, S) and (B, T ) satisfy the (CLRST ) property. Proof. Since the pair (A, S) satisfies the (CLRS ) property, there exists a sequence {xn } in X such that lim Axn = lim Sxn = z,
n→∞
n→∞
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where z ∈ S(X). As A(X) ⊂ T (X) (wherein T (X) is a closed subset of X), for each {xn } ⊂ X there corresponds a sequence {yn } ⊂ X such that Axn = T yn . Therefore, lim T yn = lim Axn = z,
n→∞
n→∞
where z ∈ S(X) ∩ T (X). Thus in all, we have Axn → z, Sxn → z and T yn → z. By (4), the sequence {Byn } converges, and in all, we need to show that Byn → z as n → ∞. On using inequality (4.1) with x = xn , y = yn , we get ψ (FAxn ,Byn (t), FSxn ,T yn (t), FAxn ,Sxn (t), FByn ,T yn (t)) ≥ 0.
(4.2)
Let Byn → l(6= z) for t > 0 as n → ∞. Then, passing to limit as n → ∞ in inequality (4.2), we have ψ (Fz,l (t), Fz,z (t), Fz,z (t), Fl,z (t)) ≥ 0, and so ψ (Fz,l (t), 1, 1, Fl,z (t)) ≥ 0, implying thereby, Fz,l (t) > 1, a contradiction. Hence z = l. Thus the pairs (A, S) and (B, T ) share the (CLRST ) property. In general, the converse of the above Lemma 4.1 is not true (see [11]). Theorem 4.1. Let A, B, S and T be self mappings of a Menger space (X, F , △) satisfying inequality (4.1) of Lemma 4.1. If the pairs (A, S) and (B, T ) share the (CLRST ) property, then (A, S) and (B, T ) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point if both the pairs (A, S) and (B, T ) are weakly compatible. Proof. Since, both the pairs (A, S) and (B, T ) share the (CLRST ) property, there exist two sequences {xn } and {yn } in X such that lim Axn = lim Sxn = lim Byn = lim T yn = z,
n→∞
n→∞
n→∞
n→∞
where z ∈ S(X)∩T (X). Since z ∈ S(X)∩T (X), there exist points u, v ∈ X such that Su = z and T v = z. Firstly, we prove that Au = z. To accomplish this, using (4.1) with x = u, y = yn , we get ψ (FAu,Byn (t), FSu,T yn (t), FAu,Su (t), FByn ,T yn (t)) ≥ 0. Taking the limit as n → ∞, we have ψ (FAu,z (t), Fz,z (t), FAu,z (t), Fz,z (t)) ≥ 0,
IMPACT OF (CLRST ) PROPERTY
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and so ψ (FAu,z (t), 1, FAu,z (t), 1) ≥ 0, yielding thereby, FAu,z (t) > 1, a contradiction, that is, Au = z. Hence z = Au = Su. Therefore u is a coincidence point of the pair (A, S). Now we assert that Bv = z. On using inequality (4.1) with x = u, y = v, we have ψ (FAu,Bv (t), FSu,T v (t), FAu,Su (t), FBv,T v (t)) ≥ 0, and so ψ (Fz,Bv (t), Fz,z (t), Fz,z (t), FBv,z (t)) ≥ 0, or ψ (Fz,Bv (t), 1, 1, FBv,z (t)) ≥ 0. implying thereby, Fz,Bv (t) > 1, a contradiction, that is, Bv = z. Hence z = Bv = T v. Therefore Bv = T v = z which shows that v is a coincidence point of the pair (B, T ). Since the pair (A, S) is weakly compatible and Au = Su = z, we deduce that Az = ASu = SAu = Sz. On using inequality (4.1) with x = z, y = v, we get ψ (FAz,Bv (t), FSz,T v (t), FAz,Sz (t), FBv,T v (t)) ≥ 0, or, equivalently, ψ (FAz,z (t), FAz,z (t), 1, 1) ≥ 0, which contradicts (ψ2 ). Hence Az = z. Therefore z = Az = Bz which shows that z is a common fixed point of the pair (A, S). Since the pair (B, T ) is weakly compatible and Bv = T v = z, we deduce that Bz = BT v = T Bv = T z. On using inequality (4.1) with x = u, y = z, we have ψ (FAu,Bz (t), FSu,T z (t), FAu,Su (t), FBz,T z (t)) ≥ 0, and so ψ (Fz,Bz (t), Fz,Bz (t), Fz,z (t), FBz,Bz (t)) ≥ 0, or, equivalently, ϕ (Fz,Bz (t), Fz,Bz (t), 1, 1) ≥ 0, which contradicts (ψ2 ). Hence Bz = z. Thus z = Bz = T z which shows that z is a common fixed point of the pair (B, T ). Therefore z is a common
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fixed point of both the pairs (A, S) and (B, T ). Uniqueness of common fixed point is an easy consequence of inequality (4.1) in view of (ψ2 ). Example 4.1. Let (X, F , △) be a Menger space, where X = [5, 15), with t-norm △ is defined by △(a, b) = min{a, b} for all a, b ∈ [0, 1] and t , if t > 0; Fx,y (t) = t+|x−y| 0, if t = 0,
for all x, y ∈ X and t > 0. Let ψ : [0, 1]4 → R as ψ(t1 , t2 , t3 , t4 ) = √ t1 − φ (min{t2 , t3 , t4 }), where φ(s) = s. Define the self mappings A, B, S and T by
A(x) =
S(x) =
5,
if x ∈ {5} ∪ (9, 15);
B(x) =
13, if x ∈ (5, 9]. if x = 5; 5, 13, x+1 2
T (x) =
if x ∈ (5, 9];
5,
10, if x ∈ (5, 9]. if x = 5; 5, 7,
, if x ∈ (9, 15).
if x ∈ {5} ∪ (9, 15);
if x ∈ (5, 9];
x − 4, if x ∈ (9, 15). 9 + n1 , {yn } = {5} or {xn } = {5},
Consider the sequences {xn } = {yn } = 9 + n1 , it is clear that both the pairs (A, S) and (B, T ) satisfy the (CLRST ) property, lim Axn = lim Sxn = lim Byn = lim T yn = 5 ∈ S(X) ∩ T (X).
n→∞
n→∞
n→∞
n→∞
Then A(X) = {5, 13} * [5, 11) = T (X) and B(X) = {5, 10} * [5, 8) = S(X). Thus, all the conditions of Theorem 4.1 are satisfied and 5 is the unique common fixed point of the pairs (A, S) and (B, T ). Here, it is worth noting that in this example S(X) and T (X) are not closed subsets of X. Also, all the involved mappings are even discontinuous at their unique common fixed point 5. Theorem 4.2. Let A, B, S and T be self mappings of a Menger space (X, F , △) satisfying the conditions (1)-(5) of Lemma 4.1. Then A, B, S and T have a unique common fixed point if both the pairs (A, S) and (B, T ) are weakly compatible. Proof. In view of Lemma 4.1, both the pairs (A, S) and (B, T ) satisfy the (CLRST ) property, that is, there exist two sequences {xn } and {yn } in X such that lim Axn = lim Sxn = lim T yn = lim Byn = z, n→∞
n→∞
n→∞
n→∞
IMPACT OF (CLRST ) PROPERTY
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where z ∈ S(X) ∩ T (X). The rest of the proof can be completed on the lines of the proof of Theorem 4.1, therefore details are omitted. Example 4.2. In the setting of Example 4.1, replace the self mappings A, B, S and T by the following (besides retaining the rest):
A(x) =
5,
if x ∈ {5} ∪ (9, 15);
10, if x ∈ (5, 9]. if x = 5; 5, S(x) = 8, if x ∈ (5, 9]; x+1 2 , if x ∈ (9, 15).
5, if x ∈ {5} ∪ (9, 15); B(x) = 6, if x ∈ (5, 9]. if x = 5; 5, T (x) = 11, if x ∈ (5, 9]; x − 4, if x ∈ (9, 15).
Clearly, both the pairs (A, S) and (B, T ) satisfy the (CLRST ) property, that is, lim Axn = lim Sxn = lim Byn = lim T yn = 1 ∈ S(X) ∩ T (X).
n→∞
n→∞
n→∞
n→∞
Notice that A(X) = {5, 10} ⊂ [5, 11] = T (X) and B(X) = {5, 6} ⊂ [5, 8] = S(X). Also the remaining conditions of Theorem 4.2 can be easily verified while 5 is the unique common fixed point of the pairs (A, S) and (B, T ). Here, it is worth noting that Theorem 4.1 can not be used in the context of this example as S(X) and T (X) are closed subsets of X. Also, all the involved mappings are even discontinuous at their unique common fixed point 5. Remark 4.1. The conclusions of Lemma 4.1 and Theorems 4.1-4.2 remain true if we replace condition (4.1) by one of the following (in view of Examples 3.3-3.8): (1) FAx,By (t) ≥ φ (min{FSx,T y (t), FAx,Sx (t), FBy,T y (t)}), where φ : [0, 1] → [0, 1] is a continuous function such that φ(s) > s for all 0 < s < 1. (2) FAx,By (t) ≥ a min{FSx,T y (t), FAx,Sx (t), FBy,T y (t)}, where a > 1. (3) FAx,By (t) ≥ aFSx,T y (t) + min{FAx,Sx (t), FBy,T y (t)}, where a > 0. (4) FAx,By (t) ≥ aFSx,T y (t) + bFAx,Sx (t) + cFBy,T y (t), where a > 1 and b, c ≥ 0(6= 1). (5) FAx,By (t) ≥ aFSx,T y (t) + b[FAx,Sx (t) + FBy,T y (t)], where a > 1 and 0 ≤ b < 1. (6) FAx,By (t) ≥ aFSx,T y (t)FAx,Sx (t)FBy,T y (t), where a > 1. Remark 4.2. Theorem 4.1 generalizes the result of Gopal et al. [8, Theorem 3.9] without any requirement of completeness (or closedness) of the space (or subspaces).
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Theorem 4.3. Let A, B, S and T be self mappings of a Menger space (X, F , △). Suppose that there exists ϕ ∈ Φ such that ϕ (FAx,By (kt), FSx,T y (t), FAx,Sx (t), FBy,T y (kt))
≥
0,
(4.3)
ϕ (FAx,By (kt), FSx,T y (t), FAx,Sx (kt), FBy,T y (t))
≥
0,
(4.4)
for all x, y ∈ X, k ∈ (0, 1) and t > 0. If the pairs (A, S) and (B, T ) share the (CLRST ) property, then (A, S) and (B, T ) have a coincidence point each. Moreover, A, B, S and T have a unique common fixed point if both the pairs (A, S) and (B, T ) are weakly compatible. Proof. The proof of this theorem is on the lines of the proof of Theorem 4.1, hence details are omitted. Example 4.3. In the setting of Example 4.1, we define ϕ(t1 , t2 , t3 , t4 ) = 14t1 − 12t2 + 6t3 − 8t4 , besides retaining the rest of the example as it stands. Then all the conditions of Theorem 4.3 are satisfied for some fixed k ∈ (0, 1) and 5 is the unique common fixed point of A, B, S and T . Remark 4.3. Theorem 4.3 improves the results of Kumar and Pant [15] and Pant and Chauhan [21] and generalizes the results of Gopal et al. [8, Theorem 3.13] without any requirement of completeness (or closedness) of the underlying space (or subspaces). By choosing A, B, S and T suitably in Theorems 4.1-4.3, one can derive corollaries involving two or three mappings. As a sample, we deduce the following natural result for a pair of self mappings by setting A = B and S = T in Theorem 4.1. Corollary 4.1. Let A and S be self mappings of a Menger space (X, F , △), where △ is a continuous t-norm. Suppose that (1) the pair (A, S) satisfies the (CLRS ) property, (2) there exists ψ ∈ Ψ such that ψ (FAx,Ay (t), FSx,Sy (t), FAx,Sx (t), FAy,Sy (t)) ≥ 0,
(4.5)
for all x, y ∈ X and t > 0. Then the pair (A, S) has a coincidence point. Moreover, A and S have a unique common fixed point if the pair (A, S) is weakly compatible.
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5. Application Now, we utilize Definition 2.12 (which is indeed a natural extension of commutativity condition to two finite families) to present a common fixed point theorem for four finite families of weakly compatible mappings in Menger space (as an application of Theorem 4.1). p q n Theorem 5.1. Let {Ai }m i=1 , {Br }r=1 , {Sk }k=1 and {Th }h=1 be four finite families of self mappings of a Menger space (X, F , △) with A = A1 A2 . . . Am , B = B1 B2 . . . Bn , S = S1 S2 . . . Sp and T = T1 T2 . . . Tq satisfying inequality (4.1) of Lemma 4.1 such that the pairs (A, S) and (B, T ) share the (CLRST ) property. Then the pair (A, S) as well as (B, T ) has a coincidence point each. p q n Moreover, {Ai }m i=1 , {Br }r=1 , {Sk }k=1 and {Th }h=1 have a unique common fixed point if the pairs of families ({Ai }, {Sk }) and ({Br }, {Th}) are commute pairwise, where i ∈ {1, 2, . . . , m}, k ∈ {1, 2, . . . , p}, r ∈ {1, 2, . . . , n} and h ∈ {1, 2, . . . , q}.
Proof. The proof follows on the lines of according to Imdad et al. [10, Theorem 3.1]. By setting A1 = A2 = . . . = Am = A, B1 = B2 = . . . = Bn = B, S1 = S2 = . . . = Sp = S and T1 = T2 = . . . = Tq = T in Theorem 5.1 we deduce the following: Corollary 5.1. Let A, B, S and T be self mappings of a Menger space (X, F , △) such that the pairs {Am , S p } and {B n , T q } share the (CLRST ) property. Suppose that there exists ψ ∈ Ψ such that ψ (FAm x,B n y (t), FS p x,T q y (t), FAm x,S p x (t), FB n y,T q y (t)) ≥ 0,
(5.1)
for all x, y ∈ X, t > 0 and m, n, p, q are fixed positive integers. Then A, B, S and T have a unique common fixed point in X if AS = SA and BT = T B. Remark 5.1. The results similar to Theorem 5.1 and Corollary 5.1 can be outlined in view of conditions (4.3)-(4.4). Acknowledgements The authors is thankful to anonymous referee for his useful suggestions to improve the first version of the manuscript. The first author is also grateful to Professor Mohammad Imdad for the reprints of his valuable papers [3, 10].
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SUNNY CHAUHAN, Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor-246 701, Uttar Pradesh, INDIA.
[email protected] ISMAT BEG, University of Central Punjab, Lahore-54770, PAKISTAN.
[email protected] B. D. PANT, Government Degree College, Champawat-262 523, Uttarakhand, INDIA.
[email protected] 
